alden johnson, teresa johnson, aimee khanmarkos/697sc/thermostatmd_fina… · alden johnson, teresa...

Thermostats in Molecular Dynamics Simulations

Alden Johnson, Teresa Johnson, Aimee Khan

University of Massachusetts Amherst

December 6th, 2012

Alden Johnson, Teresa Johnson, Aimee Khan (UMass Amherst)Thermostats in MD Simulations December 6th, 2012 1 / 29

An Interest in Thermostats

Thermostat: A modification of the Newtonian MD scheme with thepurpose of generating a statistical ensemble at a constant temperature.

Match experimental conditions

Manipulate temperatures in algorithms such as simulated annealing

Avoid energy drifts caused by accumulation of numerical errors.


Statistical Ensembles

Ensemble: a large collection of microscopically defined states of asystem, with certain constant macroscopic properties

Microcanonical (NVE)

◮ Arises in Newtonian MD simulation◮ Conserves total energy

Canonical (NVT)

◮ Implement thermostats to sample from here◮ Relevant to real behavior in experiment


Microcanonical Ensemble

Consider an isolated system

Microstate: complete description of a state of the system,microscopically

The probabilities of being in a certain microstate for themicrocanonical ensemble are uniform over all possible states


Canonical Ensemble

Follows a Gibbs distribution for probability pj of being in a givenmicrostate j with energy Ej

pj =e−βEj

Zβ, Zβ =

∑

j

e−βEj

β =1

kBT

Derivation follows from maximizing entropy


Ergodic Hypothesis

The ergodic hypothesis says the long time average of an observable f

coincides with an ensemble average of the observable 〈f 〉.

f = limt→∞

1

t

∫ t

0f (x(s)) ds

〈f 〉 =∫

Γf (x) dµ(x)

where µ is the ensemble measure and Γ is the phase space of theobservable.


Molecular Dynamics Simulation

Phase space is collection of positions q and momenta p of particles insystem

The Hamiltonian Form{

dqt = ∇pH(qt , pt) dt

dpt = −∇qH(qt , pt) dt

H(q, p) = Ekin(p) + V (q), Ekin(p) =1

2pTM−1p


Sampling from Ensembles

A canonical ensemble (constant average energy) is a distribution ofmicrocanonical ensembles (constant energy)

To sample from the canonical ensemble, the following thermostatsmodulate the energy entering and leaving the boundaries of thesystem


Velocity Rescaling

Velocities are described by Maxwell-Boltzmann distribution

P(vi ,α) =

(

m

2πkBT

)12

e−

mv2i,α2kBT

Adjust instantaneous temperature by scaling all velocities

Average Ekin per degree of freedom related to T via the equipartitiontheorem

⟨

mv2i ,α

2

⟩

=1

2kBT


Velocity Rescaling

Ensemble average → average over velocities of all particles: defineinstantaneous temperature Tc for a finite system

kBTc =1

Nf

∑

i ,α

mv2i ,α

Tc 6= T until rescaling

v ′i ,α =

√

T

Tcvi ,α


Velocity Rescaling

Disadvantages

◮ Results do not correspond to any ensemble

⋆ Does not allow the proper temperature fluctuations

◮ Localized correlation not removed

◮ Not time reversible

Advantages

◮ Straightforward to implement

◮ Good for use in warmup / initialization phase


Nose-Hoover

Based on extended Lagrangian formalism

◮ Deterministic trajectory

◮ Simulated system contains virtual variables related to real variables

⋆ Coordinates q′

i = qi

⋆ Momenta p′

i = pi/s

⋆ Time t′ =

∫ t

0dts

⋆ s: additional degree of freedom, acts as external system


Nose-Hoover

Hamiltonian given by

H =

N∑

i=1

p2i

2mi s2+ V (q) +

Qs2

2+ (3N + 1)kBT ln s

Logarithmic term required for proper time scaling: canonical ensemble

Effective mass Q associated with s◮ Determines thermostat strength◮ Q too small: system not canonical◮ Q too large: temperature control inefficient

Microcanonical dynamics on extended system give canonicalproperties


Nose-Hoover

Disadvantages

◮ Extended system not guaranteed to be ergodic

Advantages

◮ Easy to implement and use

⋆ Implement as a chain

⋆ Each link: apply thermostatting to the previous thermostat variable

◮ Increasing Q lengthens decay time of response to instantaneoustemperature jump

◮ Deterministic and time reversible


Langevin

Consider the motion of large particles through a continuum of smallerparticles

dqidt

=pi

mi

dpidt

= −δV (q)

δqi− γpi + σGi

◮ Viscous drag force proportional to velocity −γpi◮ Smaller particles give random pushes to large particle

Fluctuation-dissipation relation

σ2 = 2γmikBT


Langevin

Disadvantages

◮ Difficult to implement drag for non-spherical particles: γ related toparticle radius

◮ Momentum transfer lost: cannot compute diffusion coefficients

Advantages

◮ Damping + random force 7→ correct canonical ensemble

◮ Ergodic

◮ Can use larger time step


Andersen

Couple a system to a heat bath to impose desired temperature

Equations of motion are Hamiltonian with stochastic collision term

Strength of coupling specified by ν, the stochastic collision frequency

When particle has collision, new velocity is sampled from N (0,√T )


Andersen

Disadvantages

◮ Newtonian dynamics + stochastic collisions → Markov chain

◮ Algorithm randomly decorrelates velocity: dynamics are not physical

Advantages

◮ Allows sampling from canonical ensemble


Thermostats Summary

Thermostat Description Canonical? Stochastic?

Velocity Rescaling KE fixed to match T no no

Nose-Hoover extra degree of freedomacts as thermal reser-voir

yes no

Langevin noise and drag balanceto give correct T

yes yes

Andersen momenta occasionallyre-randomized

yes yes


Applied Math Masters Project

Molecular Dynamics Simulation

◮ Implemented in MATLAB and C++

◮ Simulations in 2 and 3 dimensions

◮ Periodic and walled boundary conditions

◮ External fields such as gravity

◮ Optimization⋆ OpenMP

⋆ CUDA


Lennard-Jones Potential


Numeric Integration

Verlet Algorithm

pn+1/2 = pn − ∆t

2∇V (qn)

qn+1 = qn +∆tM−1pn+1/2

pn+1 = pn+1/2 − ∆t

2∇V (qn+1)

◮ Preserves modified Hamiltonian


Thermostat Implementation

Velocity Rescaling

◮ pi →√

TTcpi

Anderson◮ ν = 1%, 0.1%, 0.05%◮ Velocities are sampled from N (0,

√T )

Langevin◮ BBK algorithm

pn+1/2 = pn − ∆t2 ∇V (qn)− ∆t

2 γ(qn)M−1pn +√

∆t2 σ(qn)G n

qn+1 = qn +∆tM−1pn+1/2

pn+1 = pn+1/2−∆t

2∇V (qn+1)−∆t

2γ(qn+1)M−1pn+1+

√

∆t

2σ(qn+1)G n+1

◮ γ chosen to be constant◮ σ =

√2γMkBT

◮ G sampled from a standard normal distribution


Velocity Rescaling - Instantaneous Temperature

0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5T

emp

time step


Anderson - Instantaneous Temperature

0 2000 4000 6000 8000 100000

1

2

3

4

5

6

Tem

p

time step

ν = 1%ν = 0.1%ν = 0.05%


Anderson - Histogram of Velocity Distribution

−10 −5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

velocity

p(v)


Langevin Instantaneous Temperature

0 2 4 6 8 10

x 104

1

2

3

4

5

6

7

Tem

p

time step

γ = 0.1γ = 0.01γ = 0.001


Langevin - Histogram of Velocity Distribution

−10 −5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2p(

v)

velocity


References

Barsegov, Valeri (2007)

Different ensembles in molecular dynamics simulations

http://faculty.uml.edu/vbarsegov/teaching/bioinformatics/lectures/MDEnsemblesModified.pdf

Carloni, Paolo (2009)

Simulation of Biomolecules

http://www.grs-sim.de/cms/upload/Carloni/Presentations/Strodel3.pdf

Hunenberger, Philippe H. (2005)

Thermostat Algorithms for Molecular Dynamics Simulations

Adv. Polym. Sci. 173:105-149

Tuckerman, Mark E.(2010)

Statistical Mechanics: Theory and Molecular Simulation

Oxford Graduate Texts

Zhao, Yanxiang (2011)

Brief introduction to the thermostats

http://www.math.ucsd.edu/ y1zhao/ResearchNotes/ResearchNote007Thermostat.pdf


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